Optimizing Supply Chain Inventory: A Mixed Integer Linear Programming Approach

Abstract

Abstract: Inventory supply chain planning involves determining the quantity of products to be transported among entities within a specified planning horizon. Often, inventory levels are reviewed at set intervals: (1) Background: In this paper, periodic review (s,S) policy is used to optimize inventories from an integrated perspective of inventory management across the supply chain. The decision to place an order and the order quantity are based on the inventory level at the review time. If the inventory falls below a certain level (s), an order is placed to replenish it to a target level (S); (2) Methods: The planning model is implemented using a mixed integer linear programming model. It determines the inventory levels, supply levels and the (s) and (S) levels for each entity, as well as the flow of products between them. To test the model, a case study is conducted to demonstrate its applicability; (3) Results: The experimental data confirm the model’s validity, as its behavior aligns with the expectations for a periodic review (s,S) policy; (4) Conclusions: Since a fixed replenishment frequency is mandatory, with no continuous inventory review required, this policy offers simplicity and ease of implementation, making it a practical choice for certain inventory management.

1. Introduction

Firms are focused on coordinating the movement of goods throughout the supply chain to fulfill client needs while minimizing costs. Inventory planning plays a vital role in these systems, as it helps balance costs and meet customer expectations by maintaining a steady supply–demand equilibrium. In fact, there are several challenges, for example, fluctuating demand, supply disruptions, cost pressures and different inventory review policies used by supply chain members.

In this scenario, inventory planning is a critical and complex task, where decisions by one member of the chain can have varying effects on the other entities involved. This problem attracted attention when a Mixed Integer Linear Programming (MILP) model was introduced to address inventory planning across the chain [1]. Strategies for coordinating movements across the supply chain were explored, showing their influence on the planning of inventory and supply plans of individual participants [2].

Some authors noted that traditionally, firms only exchange order information, but advancements in information technology now enable the quick and inexpensive sharing of demand and inventory data. Numerical studies demonstrated that average supply chain costs were lower when shared information was implemented [3]. These findings were corroborated by other authors, who examined how integrating and coordinating inventory policies among different supply chain participants can streamline material flow, reduce costs, and better meet customer demand [4].

Optimizing inventory within a single echelon often leads to surplus inventory without clear benefits to service level. Researchers suggested a new approach for supply chain inventory planning, emphasizing the importance of considering systems with many stages, which was lacking in the related literature. The methodology provided a valuable tool for managing supply chain systems [5]. Later on, work on inventory models in a one-to-many vendor–buyer network was conducted. The authors found that fostering relationships through collaboration improves inventory control performance [6].

These studies underscore the need and challenges of creating detailed models capable of handling diverse and complex system topologies while incorporating inventory planning. These models can support decision-making processes in managing goods within intricate network systems. However, addressing an integrated view of inventory along the network is still lacking in the current literature. The integration within the supply chain related to inventory review policy is challenging due to the decisions that can be made by each chain member.

This manuscript discusses the periodic review policy for multiple-echelon supply chain inventory management, a topic that is underrepresented in the current literature. Furthermore, one of the key insights is the mathematical model, which differs from existing models in the literature. Therefore, the scope of this manuscript is to develop a supply chain planning model using an inventory periodic review (s,S) policy, which can guide decision-makers in determining the transportation of goods based on an optimization approach. The model could be applied to different supply chain topologies with two stages. Costs include shipping, carrying, and ordering.

Summing up, the main contributions of this paper to the literature are as follows:

• A supply chain planning model that takes into account supply and inventory levels;
• Short-term planning of distribution process networks;
• The combination of inventory planning with inventory policy, which enables better operational decisions (e.g., minimizing ordering, holding, and transportation costs) for supply chain managers.

The following sections of the manuscript are organized as follows. The literature review is presented in Section 2. Section 3 outlines the model using periodic review policy. Section 4 presents and solves a case study. Finally, conclusions and future research directions are in Section 5.

2. Literature Review

Inventory management in a multiple stage supply chain is more complex than in a single stage due to the relationships between entities. These complexities lie on lead time variations, demand propagation, or coordination challenges, such as inventory review policies. Some authors have highlighted this challenge by discussing the inventory management features throughout supply chains [7,8].

Since the policy discussed in this paper is the periodic review (s,S) policy, inventory is checked at regular intervals. Upon its level falling below (s), an order is initiated to restore the inventory to (S). Various methods in related literature, simulation-based optimization, and heuristics have been proposed to determine the policy parameters [9,10]. It was emphasized that developing effective inventory management policies is crucial, as successful inventory management depends on having efficient methods to plan the inventory needs across the network. Researchers introduced new viewpoints on collaborative inventory management, pointing out that inventory challenges frequently result from inadequate management of supply chain processes. Therefore, effective inventory management requires collaboration and integration among all supply chain entities to enhance efficiency [11].

While inventory enhances flexibility, it also generates costs, making it essential to strike the right balance between them [12]. Literature on managing the inventory within networks has begun to address these issues, exploring various policies and comparing inventory review approaches [13]. Using a communication mode that fosters the exchange of information between members in different echelons in the supply chain yields a higher service level and reduced cycle time when compared to a scenario where the members in different echelons act independently. Some authors presented an approach to discover inventory parameters on a system with a warehouse and several retailers where an optimization of reorder points for a review policy was considered in a two-stage supply chain [14]. Other authors presented a method for efficiently transporting products through a supply chain over a specified planning time horizon. Their approach focused on determining the optimal distribution requirements plan [15].

In traditional deterministic supply chain modeling, demand is considered certain. There are no lost sales, and lead times are not taken into account. Retailers place orders to meet customer demands, which in turn creates demand at the warehouses. Costs at each supply chain entity include ordering and holding costs [16,17]. Some authors created a model for inventory planning on a supply chain with a warehouse and several retailers. They used a MILP model to determine the optimal plan that would minimize the associated costs [18]. Other authors researched a multi-echelon network with two centers of distribution and two retailers, developing a decision rule to minimize the expected costs of outstanding orders, assuming synchronized decision-making among retailers [19]. Some authors proposed a model with lateral transshipment, where retailers use a periodic review policy [20]. Authors reviewed models integrating location with operational management, while other authors tackled the inventory planning problem by proposing policies for inventory planning in process networks [21,22]. Special attention was given to mathematical models, however the use of periodic inventory review policy in supply chains was not addressed.

More recently, authors researched the location of inventory problems using optimization through a muti-period time horizon [23], and others developed a mixed integer two-stage inventory problem. The problem is a combination of two review policies [24]. Some authors also addressed supply-chain planning including continuous and periodic review models [25]. Other authors proposed a model that optimizes the parameters in a multipl-stage network [26]. Authors studied a mixed integer programming model integrating key location allocation and inventory replenishment decisions [27]. Multi-period modeling is often neglected in those studies, which is far from what happens in practice [28,29]. Therefore, these features are lacking in the literature, which mostly focuses on single-period systems. Recent research has emphasized applying these optimization methods to model inventory planning in supply chains. However, inventory management is often discussed without detailed supply chain review policies as highlighted [30,31]. Moreover, several studies in the literature suggest that it is essential to conduct more in-depth research on supply chain management to address practical challenges effectively [32]. This manuscript fills this gap by applying the periodic review (s,S) policy to the problem of inventory planning in the supply chain, providing unique insights as this policy offers simplicity and ease of implementation, making it a practical choice for certain inventory management scenarios.

Based on this analysis, there is still potential to generalize inventory planning and policy models by exploring system structures and operations. Therefore, a closer examination of real-world problems is necessary, particularly considering inventory system parameters such as supplies, inventories, and transportation flows across multi-stage networks.

This manuscript develops a model that determines inventories, orders, and shipping plans for retailers and warehouses, with the aim of minimizing distribution supply chain costs. This planning tool, designed to better reflect real-world scenarios, employs the periodic review (s,S) policy adopted by each supply chain entity.

3. Periodic Review (s,S) Policy under Supply Chain Problem

The supply chain inventory planning problem using a review policy is formulated as a MILP model. This system incorporates two stages, each potentially consisting of multiple entities. Product flows can occur between any two entities across these stages, and the entire system is supported by a central warehouse. Retailers place orders to meet local demand, and to fulfill these orders, one or more flows from various entities are planned, with the combined flows satisfying the retailer’s order.

To capture all decision-making processes effectively, the model is based on a discrete time, where time intervals have a fixed duration for the time scale. Thus, the mathematical formulation is presented.

3.1. Indices

i, j, k    entity

t   period

3.2. Sets

$$i, j,k\in I=\left\{0, 1, 2,\dots ,NW,NW+1,NW+2,\dots ,NW+NR\right\}$$

$$t\in T=\left\{1, 2, \dots ,NT\right\}$$

$$W_o=\left\{0\right\}$$

$$W=\left\{1, 2, \dots ,NW\right\}$$

$$R=\left\{1,2,\dots ,NR\right\}$$

3.3. Parameters

$$D_{kt}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{demand\; at\; entity\; k\; in\; period\; t}$$

$$H{C}_j\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{holding\; cost\; per\; unit\; at\; entity\; j\; per\; period}$$

$$It{o}_j\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{initial\; inventory\; level\; at\; entity\; j}$$

$$L_{jk}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{lead\; time\; of\; order\; from\; entity\; j\; to\; entity\; k}$$

$$M\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{big-M}$$

$$O{C}_j\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{cost\; per\; order\; at\; entity\; j}$$

$$T{C}_{jk}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Shipping\; cost\; per\; unit\; from\; entity\; j\; to\; entity\; k}$$

3.4. Continuous Variables

$$IN{V}_{jt}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{inventory\; at\; entity\; j\; at\; the\; end\; of\; the\; period\; t}$$

$$A{P}_{jkt}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{amount\; of\; product\; from\; entity\; j\; to\; entity\; k\; during\; period\; t}$$

$$s_j\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{level\; s\; at\; entity\; j}$$

$$s{S}_j\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{level\; S\; at\; entity\; j}$$

3.5. Binary Variables

$$x{a}_{jt}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{1\; if\; replenishment\; is\; allowed\; at\; entity\; j\; in\; period\; t,\; 0\; if\; not}$$

$$x{p}_{jn}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{1\; if\; replenishment\; profile\; n\; is\; chosen\; at\; entity\; j,\; 0\; if\; not}$$

$$x{r}_{jt}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{1\; if\; replenishment\; is\; required\; at\; entity\; j\; in\; period\; t,\; 0\; if\; not}$$

$$y{r}_{jt}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{1\; if\; replenishment\; is\; actually\; performed\; at\; entity\; j\; in\; period\; t,\; 0\; if\; not}$$

3.6. Mathematical Model

The objective function is as follows.

$$\mathrm{Minimize} \mathrm{total} \cos \mathrm{t}={\displaystyle \sum }_{j\in W}{\displaystyle \sum }_{t\in T}O{C}_j\times y{r}_{jt}+{\displaystyle \sum }_{k\in R}{\displaystyle \sum }_{t\in T}O{C}_k\times y{r}_{kt}+{\displaystyle \sum }_{j\in W}{\displaystyle \sum }_{t\in T}H{C}_j\times IN{V}_{jt}$$

$${\displaystyle \sum }_{k\in R}{\displaystyle \sum }_{t\in T}H{C}_k\times IN{V}_{kt}+{\displaystyle \sum }_{i\in W_0}{\displaystyle \sum }_{j\in W}{\displaystyle \sum }_{t\in T}T{C}_{ij}\times A{P}_{ijt }+{\displaystyle \sum }_{j\in W}{\displaystyle \sum }_{k\in R}{\displaystyle \sum }_{t\in T}T{C}_{jk}\times A{P}_{jkt}$$

(1)

Subject to the following constraints:

$$IN{V}_{jt}=It{o}_j+A{P}_{i,j,t-L_{ij}|L_{ij}=0}-{\displaystyle \sum }_{k\in R}A{P}_{jkt}, i\in W_0,j\in W,t=1$$

(2)

$$IN{V}_{jt}=IN{V}_{j,t-1}+A{P}_{i,j,t-L_{ij}|L_{ij}<t}-{\displaystyle \sum }_{k\in R}A{P}_{jkt}, i\in W_0, j\in W,t\in T\backslash \left\{1\right\}$$

(3)

$$IN{V}_{jt}=It{o}_j, j\in W,t=NT$$

(4)

$$IN{V}_{kt}=It{o}_k+{\displaystyle \sum }_{j\in W}A{P}_{j,k,t-L_{jk}|L_{jk}=0}-D_{kt}, k\in R,t=1$$

(5)

$$IN{V}_{kt}=IN{V}_{k,t-1}+{\displaystyle \sum }_{j\in W}A{P}_{j,k,t-L_{jk}|L_{jk}<t}-D_{kt}, k\in R,t\in T\backslash \left\{1\right\}$$

(6)

$$IN{V}_{kt}=It{o}_k, k\in R,t=NT$$

(7)

$${\displaystyle \sum }_{n\in N_t}x{p}_{jn}=x{a}_{jt}, j\in W,t\in T$$

(8)

$$A{P}_{ijt}\le M\times x{a}_{jt}, i\in W_0,j\in W,t\in T$$

(9)

$${\displaystyle \sum }_{n\in N_t}x{p}_{jn}=1, j\in W$$

(10)

$$-M\times x{r}_{jt}+0.0001\le IN{V}_{jt}-s_j, j\in W,t\in T$$

(11)

$$IN{V}_{jt}-s_j\le M\times \left(1-x{r}_{jt}\right), j\in W,t\in T$$

(12)

$$x{a}_{jt}+x{r}_{jt}\le y{r}_{jt}+1, j\in W,t\in T$$

(13)

$$y{r}_{jt}\le x{a}_{jt}, j\in W,t\in T$$

(14)

$$y{r}_{jt}\le x{r}_{jt}, j\in W,t\in T$$

(15)

$$A{P}_{ijt}-s{S}_j+IN{V}_{jt}\le M\times \left(1-y{r}_{jt}\right), i\in W_0,j\in W,t\in T$$

(16)

$$-A{P}_{ijt}+s{S}_j-IN{V}_{jt}\le M\times \left(1-y{r}_{jt}\right), i\in W_0,j\in W,t\in T$$

(17)

$$A{P}_{ijt}\le M\times y{r}_{jt}, i\in W_0,j\in W,t\in T$$

(18)

$${\displaystyle \sum }_{n\in N_t}x{p}_{kn}=x{a}_{kt}, k\in R,t\in T$$

(19)

$${\displaystyle \sum }_{j\in W}A{P}_{jkt}\le M\times x{a}_{jt}, k\in R,t\in T$$

(20)

$${\displaystyle \sum }_{n\in N_t}x{p}_{kn}=1, k\in R$$

(21)

$$-M\times x{r}_{kt}+0.0001\le IN{V}_{kt}-s_k, k\in R,t\in T$$

(22)

$$IN{V}_{kt}-s_k\le M\times \left(1-x{r}_{kt}\right), k\in R,t\in T$$

(23)

$$x{a}_{kt}+x{r}_{kt}\le y{r}_{kt}+1, k\in R,t\in T$$

(24)

$$y{r}_{kt}\le x{a}_{kt}, k\in R,t\in T$$

(25)

$$y{r}_{kt}\le x{r}_{kt}, k\in R,t\in T$$

(26)

$${\displaystyle \sum }_{j\in W}A{P}_{jkt}-s{S}_k+IN{V}_{kt}\le M\times \left(1-y{r}_{kt}\right), k\in R,t\in T$$

(27)

$$-{\displaystyle \sum }_{j\in W}A{P}_{jkt}+s{S}_k-IN{V}_{kt}\le M\times \left(1-y{r}_{kt}\right), k\in R,t\in T$$

(28)

$${\displaystyle \sum }_{j\in W}A{P}_{jkt}\le M\times y{r}_{jt}, k\in R,t\in T$$

(29)

$$IN{V}_{jt},A{P}_{jkt},s_j,s{S}_j\ge 0, j\in I, k\in I,t\in T$$

(30)

$$x{a}_{jt}, x{p}_{jt},x{r}_{jt}, y{r}_{jt}\in \left\{0,1\right\} j\in I, t\in T$$

(31)

The purpose is to minimize the cost using the objective function (1), which comprises the following: the ordering cost for warehouses and retailers (first two terms); the carrying cost at both echelons (next two terms); and, lastly, the transportation cost from the central warehouse to warehouses and from warehouses to retailers (fourth and fifth terms). The practical significance and the basis for the value of each cost coefficient is as follows: (i) ordering cost is the cost to place an order, independent of the quantity; (ii) holding cost is the cost per unit of inventory on hand; (iii) transportation cost is the cost per unit of shipping from an entity to other (see Section 3.3).

Constraints (2) and (3) specify the amount entered at each warehouse that equals the amount of product from the central warehouse per period, accounting for order lead time. The total out amount at each warehouse matches the sum of the amounts of product to the retailers per period. For the first period, the inventory at the end is determined by constraint (2), which considers the beginning inventory level. For subsequent periods, the inventory at the end of each period is governed by constraint (3).

A similar analysis can be performed for each retailer based on constraints (5) and (6).

To minimize the influence of the initial inventory on the subsequent planning horizon, both the beginning inventory (for period 1) and the final inventory (for period NT) are set to be equal, as indicated in constraints (4) and (7).

The periodic review (s,S) policy is defined with the following constraints. For warehouses it is as follows: According to constraint (8), the option of a specific profile determines if replenishing is permitted in a given period, where ($N_t$) represents the set of profiles permitting replenishment in period (t). The flow from the central warehouse to a warehouse (j) is permitted only if the binary variable ($x{a}_{jt}$) equals 1, where (M) is a big M (constraint (9)). Constraint (10) says that only one replenishment profile can be selected. After setting the constraints to allow replenishing on the periods permitted by the chosen profile, there is a need to enforce that replenishment occurs when the inventory level falls below the inventory position (s). Therefore, constraints (11) and (12), are defined. A replenishment will occur when it is both required and permitted, stated by constraints (13)–(15). Once a replenishing choice is determined, the quantity must bring back the inventory position to (S). Constraints (16)–(18) illustrate this logic. If the shipping quantity is non-zero, the binary variable ($y{r}_{jt}$) equals 1, as indicated in constraint (18).

Similar constraints for retailers are specified in constraints (19)–(29).

The practical implications of these previous variables (decision variables) for supply chain managers are to know when and how much to order using the periodic review policy within supply chain inventory planning.

As previously defined, the model employs both continuous and binary variables (30)–(31).

The model consists of objective function (1) subject to constraints (2) to (31).

4. Case Study

To demonstrate the applicability of the model presented in the previous section, a retail company case study using the proposed review policy is analyzed. The problem was solved using the GAMS 41 modeling language on an Intel CORE i7 CPU 3.34 GHz and 16 GB RAM.

4.1. Data

A deterministic set of data is assumed. Due to confidentiality reasons, the data provided by the retail company has been changed but still describes the real operation. The network structure is as follows: (1) one central warehouse; (2) two warehouses; (3) four retailers. A planning horizon for 15 periods was assumed, since it is the planning horizon usually utilized by the company. The ordering cost for warehouses and retailers is 30 €. The holding costs are 0.2 € and 0.6 € per period, respectively, for warehouses and retailers. Parameters are presented on

Table 1. Unit transportation cost from each warehouse to each retailer (in EUR).

	Warehouse 1	Warehouse 2	Retailer 1	Retailer 2	Retailer 3	Retailer 4
Warehouse 0	0.55	0.22	-	-	-	-
Warehouse 1	-	-	0.22	0.20	0.32	0.38
Warehouse 2	-	-	0.68	0.52	0.34	0.1

,

Table 2. Initial inventory level for each warehouse and each retailer (in units).

Warehouse 1	Warehouse 2	Retailer 1	Retailer 2	Retailer 3	Retailer 4
1200	1100	350	450	500	600

,

Table 3. Demand at each retailer per period (in units).

Retailer 1	Retailer 2	Retailer 3	Retailer 4
50	60	70	80

and

Table 4. Order lead time between supply chain entities (in periods).

	Warehouse 1	Warehouse 2	Retailer 1	Retailer 2	Retailer 3	Retailer 4
Warehouse 0	3	3	-	-	-	-
Warehouse 1	-	-	3	3	3	3
Warehouse 2	-	-	3	3	3	3

.

Table 5. Replenishment profiles under periodic review policy for 15-period planning horizon.

Profile	t1	t2	t3	t4	t5	t6	t7	t8	t9	t10	t11	t12	t13	t14	t15
#1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
#2	1	0	1	0	1	0	1	0	1	0	1	0	1	0	1
#3	0	1	0	1	0	1	0	1	0	1	0	1	0	1	0
#4	1	0	0	1	0	0	1	0	0	1	0	0	1	0	0
#5	0	1	0	0	1	0	0	1	0	0	1	0	0	1	0
#6	0	0	1	0	0	1	0	0	1	0	0	1	0	0	1
#7	1	0	0	0	1	0	0	0	1	0	0	0	1	0	0
#8	0	1	0	0	0	1	0	0	0	1	0	0	0	1	0
#9	0	0	1	0	0	0	1	0	0	0	1	0	0	0	1
#10	0	0	0	1	0	0	0	1	0	0	0	1	0	0	0
#11	1	0	0	0	0	1	0	0	0	0	1	0	0	0	0
#12	0	1	0	0	0	0	1	0	0	0	0	1	0	0	0
#13	0	0	1	0	0	0	0	1	0	0	0	0	1	0	0
#14	0	0	0	1	0	0	0	0	1	0	0	0	0	1	0
#15	0	0	0	0	1	0	0	0	0	1	0	0	0	0	1

shows the replenishment profiles for a 15-period planning horizon. Each replenishment profile specifies the periods on which replenishment is permitted (marked with 1).

4.2. Experimental Results

The optimal inventory supply chain planning for the 15-period horizon indicates the material amount shipped among entities. Results show that costs are minimized when warehouse 1 supplies retailers 1, 2 and 3, and warehouse 2 supplies retailers 2, 3 and 4 (see Appendix A).

The figures below detail the results obtained for the particular case of warehouse 1 and retailers 1, 2 and 3. Figure 1 shows the inventory level and order amount for retailer 1. The profile of replenishment selected (profile #10) indicates that the inventory needs to be checked every 4 days (review frequency) and starts on day 4 (starting day). The replenishment profile is a binary variable (decision variable), as presented in the mathematical formulation (Section 3). Therefore, the optimal solution of the optimization model is profile #10. This selected profile has advantages over other profiles, since it contributes to the minimum cost. When inventory levels drop by 250 units (s level), an order is placed to bring the inventory level back to 500 units (S level). As can be seen, retailer 1 places a first order amount of 350 units, and a second of 400 units, as the inventory level at period 4 is higher than at period 12. The replenishment takes three days (lead time), from day 4 to day 7 and a new replenishment repeats again from day 12 to day 15. A similar analysis is performed in Figure 2 and Figure 3 for retailers 2 and 3, respectively.

Figure 4 shows the inventory level and order amount for warehouse 1. The profile of replenishment selected (profile #6) specifies that inventory should be reviewed every 3 days (review frequency) and start on day 3 (starting day). As can be seen, warehouse 1 places a first order amount of 720 units and a second of 980 units when the inventory level drops below 480 units (s level). The replenishment takes three days (lead time), from day 9 to day 12 and a new replenishment repeats again from day 12 to day 15.

Table 6. Policy parameters.

	s Level	S Level	Replenishment Profile
Warehouse 1	480	1200	#6
Warehouse 2	0	1100	#1
Retailer 1	250	500	#10
Retailer 2	270	630	#10
Retailer 3	360	710	#10
Retailer 4	440	840	#10

outlines the policy parameters for all entities. As the daily demand rate increases from retailer 1 to retailer 4 (see Table 3), an increase trend can be observed for replenishment levels too.

Table 7. Computational statistics.

MIP Solution (Euro)	14 811
Best possible (euro)	14 811
Relative gap (%)	0
Single equations	1 003
Single variables	628
Discrete variables	360
Computational time (second)	128.80

presents the computational statistics. The solution of the model cost 14 811 EUR. An optimal solution (0\% of relative gap) in about 2 min was achieved.

4.3. Discussion of the Results

This case study examines a policy where inventory is reviewed at specific intervals, known as the periodic review (s,S) policy. Determining the parameters for this policy can be challenging, with several methods, such as heuristics and simulation-based optimization, discussed in the literature. Mathematical programming models (as is the case with the model in this research) are called exact models in which an optimal solution is obtained, however they can become a major computational challenge. The other methods mentioned normally do not pose difficult computational challenges, but the solution found may not be optimal. However, this work proposes a MILP model, which provides a good alternative for practical applications.

The experimental results presented in the previous subsection confirm the model’s validity, as its behavior aligns with the expectations for a periodic review (s,S) policy. Inventory checks are conducted on scheduled days, and the decision-making process for orders and quantities depends on the inventory level at the time. If the inventory level drops below a predetermined threshold (s), an order is triggered to replenish it to the target level (S). This means ordering ($s{S}_j$—$IN{V}_{jt}$) units. The model determines inventory levels, order amounts, (s), and (S) level points. These inventory and order parameters are crucial to the supply chain operations manager’s decision-making. In this manuscript, the lead time (parameter) is given by the retail company, and it is certain and known. Stockouts are avoided, since demand and lead time are not uncertain. Operational costs are minimized as a key objective.

The inventory volume increases from retailer 1 to retailer 4. This happens because daily demand also increases from retailer 1 to retailer 4. It means that inventory level (decision variable) is compatible with demand level in each retailer. Furthermore, the (s) and (S) level points also increase to meet demand satisfaction during the lead time.

The company employs a fixed mandatory replenishment frequency, without the need for continuous inventory review. This policy offers simplicity and ease of implementation, making it a practical choice for the retail company in certain inventory management scenarios, however it may incur undesirable stockouts. The inclusion of the same inventory review policy (periodic review) for all members of the supply chain made it applicable for real-world use.

Within the planning horizon of 15 periods, two cycles of inventory replenishment can be seen (for example see Figure 1). Note that, for a lead time value greater than that used by the case study company (3 periods), a longer planning horizon is recommended in order to capture at least two inventory replenishment cycles.

5. Conclusions

This work introduces inventory planning, and a review policy model tailored to the supply chain management. The modulation purpose is to reduce costs by providing an optimized inventory and distribution plan. It employs the periodic review (s,S) policy for inventory planning, determining inventory levels, order quantities, minimum stock levels (s), maximum stock levels (S), and shipping quantities between supply chain facilities.

The proposed model enhances the supply chain inventory management literature, not only due to the limited number of studies on this topic, but also because of the lack of research extending this planning to a multi-period, multi-warehouse supply chain using a MILP model.

The results indicate that warehouses and retailers can implement this inventory policy within a supply chain, collaborating to achieve the lowest operational costs. The key issue is the use of the same inventory review policy (periodic review) for all chain members. The mathematical model fosters data sharing and converging objectives, improving performance of the supply chain. From the presented work, a set of managerial insights and practical implications can be derived:

• Including an inventory review policy in the supply chain planning model makes it applicable for real-world use;
• Where a fixed replenishment frequency is mandatory, with no continuous inventory review required, this policy offers simplicity and ease of implementation, making it a practical choice for certain inventory management scenarios;
• Integrating inventory planning with inventory review policy effectively combines tactical with operational decision-making.

There are several limitations that need to be addressed, particularly more comprehensive validation of the proposed model. It is important to study more complex supply chain topologies over longer durations and with a greater variety of products. The mathematical model, in current formulation, does not address the uncertainty of demand and lead times (a new mathematical formulation is needed under stochastic programming).

Future research on the mathematical formulation of the model under different industry characteristics (e.g., manufacturing, retailing, etc.) could be encouraged. Another research direction of the study could be the adaptation of the model for more than three echelons of the supply chain, including different types of cost for multi-modal transportation. Another direction for future research could be the substitutability and complementarity of different products. Additionally, other aspects of inventory planning systems, such as defining safety stock levels and comparing different inventory policies under uncertain conditions, should be further investigated.